A wavelet approach to modelling the evolutionary dynamics across ordered replicate time series

Experimental time series data collected across a sequence of ordered replicates often crop up in many fields, from neuroscience to circadian biology. In practice, it is natural to observe variability across time in the dynamics of the underlying process within a single replicate and wavelets are essential in analysing nonstationary behaviour. Additionally, signals generated within an experiment may also exhibit evolution across replicates even for identical stimuli.

We propose the Replicate-Evolving Locally Stationary Wavelet process (REv-LSW) which gives a stochastic wavelet representation of the replicate time series. REv-LSW yields a natural desired time- and replicate-localisation of the process dynamics, capturing nonstationary behaviour both within and across replicates, while accounting for between-replicate correlation.
Firstly, we rigorously develop the associated wavelet spectral estimation framework along with its asymptotic properties for the particular case that replicates are uncorrelated. Next, we crucially develop the framework to allow for dependence between replicates. By means of thorough simulation studies, we demonstrate the theoretical estimator properties hold in practice.

Finally, it is unreasonable to make the typical assumption that all replicates stem from the same process if a replicate spectral evolution exists. Thus, we propose two novel tests that assess whether a significant replicate-effect is manifest across the replicate time series. Our modelling framework uses wavelet multiscale constructions that mitigate against the potential nonstationarities, across both times and replicates. Thorough simulation studies prove both tests to be flexible tools and allow the analyst to accordingly tune their subsequent analysis.

Throughout this thesis, our work is motivated by an investigation into the evolutionary dynamics of brain processes during an associative learning experiment. The neuroscience data analysis illustrates the utility of our proposed methodologies and demonstrates the wider experimental data analysis achievable that is also of benefit to other experimental fields, e.g. circadian biology, and not just the neurosciences.